On Digraph Width Measures

in Parameterized Algorithmics

(extended abstract)

Robert Ganian1, Petr Hlineny1, Joachim Kneis2, Alexander Langer2,Jan Obdrzalek1, and Peter Rossmanith2

1 Faculty of Informatics, Masaryk University, Brno, Czech Republic{xganian1,hlineny,obdrzalek}@fi.muni.cz

2 Theoretical Computer Science, RWTH Aachen University, Germany{kneis,langer,rossmani}@cs.rwth-aachen.de

Abstract. In contrast to undirected width measures (such as tree-width or clique-width), which have provided many important algo-rithmic applications, analogous measures for digraphs such as DAG-width or Kelly-width do not seem so successful. Several recent papers,e.g. those of Kreutzer–Ordyniak, Dankelmann–Gutin–Kim, or Lampis–Kaouri–Mitsou, have given some evidence for this. We support this di-rection by showing that many quite different problems remain hard evenon graph classes that are restricted very beyond simply having smallDAG-width. To this end, we introduce new measures K-width and DAG-depth. On the positive side, we also note that taking Kante’s directedgeneralization of rank-width as a parameter makes many problems fixedparameter tractable.

1 Introduction

The very successful concept of graph tree-width was introduced in the contextof the Graph Minors project by Robertson and Seymour [RS86,RS91], and itturned out to be very useful for efficiently solving graph problems. Tree-widthis a property of undirected graphs. In this paper we will be interested in directedgraphs or digraphs.

Naturally, a width measure specifically tailored to digraphs with all the niceproperties of tree-width would be tremendously useful. The properties of such ameasure should include at least the following:

i) The width measure is small on many interesting instances.ii) Many hard problems become easy if the width measure is bounded.

Obviously, there is a conflict between these goals, and consequently we can expectsome trade-off. On the search for such a digraph measure, several suggestionswere made, starting with directed tree-width [JRST01], and being complementedrecently with several new approaches including DAG-width [Obd06,BDHK06],Kelly-width [HK08], entanglement [BG04], D-width [Saf05], directed path-width [Bar06] (defined by Reed, Seymour, and Thomas), and —although quitedifferent —bi-rank-width [Kan08] (see Section 2).

Some positive results were encouraging: The Hamiltonian path problem canbe solved in polynomial time (XP) if the directed tree width, the DAG-width,or the Kelly-width are bounded by a constant [JRST01]. More recently, it hasbeen shown that parity games can be solved in polynomial time on digraphs ofbounded DAG-width [BDHK06] and Kelly-width [HK08].

Are more results just waiting around the corner and do we just have to waituntil we get more familiar with these digraph measures? It is the aim of thispaper to answer this question, at least partially.

Unfortunately, as encouraging as the first positive results are, there is alsothe negative side. Hamiltonian path is W[2]-hard on digraphs of bounded DAG-width [LKM08], and some other natural problems even remain NP-hard on di-graphs of low widths [KO08,DGK08,LKM08]. One of the main goals of this paperis to show that not only many problems are hard on DAGs, but rather that theyremain hard even if we very severely further restrict the graphs structure.

We introduce two digraph measures for this purpose: K-width and DAG-depth. While K-width (Section 2.3) restricts the number of different simple pathsbetween pairs of vertices, DAG-depth (Definition 2.6) is the directed analogof tree-depth [NdM06]. K-width and DAG-depth are very restrictive digraphmeasures; at least as high as DAG-width, and often much higher.

The problems we consider in this paper (and formally define in Section 3) areHamiltonian path (HAM), Disjoint paths (k-Path), Directed Dominating Set(DiDS), unit cost Directed Steiner Tree (DiSTP), Directed Feedback Vertex Set(DFVS), Kernel (Kernel), Maximum Directed Cut (MaxDiCut), OrientedColouring (OCN), MSO1 model checking (φ-MSO1mc), solving Parity Games(Parity) and LTL-model checking (φ-LTLmc). See Table 1 in Section 3.

It turns out that most of the aforementioned problems are not only hard forDAG-width, but even for constant K-width and DAG-depth, or on DAGs. Thiscan be seen as a strong indication that DAG-width or related measures are notyet the right parameters for dealing with standard digraph problems.

On the other hand, one width measure that fares much better in Table 1is bi-rank-width (Definition 2.4), a width measure generalizing the rank-widthof undirected graphs [Kan08]. Nearly all of our problems are fixed parametertractable or at least in XP with respect to this parameter. Even better, unlikeas for DAG-width or Kelly-width, finding an optimal bi-rank-decomposition isknown to be in FPT [HO08,Kan08].

2 Digraph Width Measures

The first wave of directed measures to appear shared the following features:

i) On bidirected orientations of graphs they coincided with the tree-width.ii) These measures were strongly based on some variant of the directed cops-

and-robber game on a digraph: There are k cops and a robber. Each copcan either occupy a vertex, or move around in a helicopter, and the robberoccupies a vertex. The robber can, however, see the helicopter landing, and

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can move at a great speed along a cop-free directed path to another vertex.The objective of the cops is to capture the robber by landing on the vertexcurrently occupied by him, the objective of the robber is to avoid capture.

iii) Point (ii) implied that DAGs and other graphs where vertices could be or-dered in such a way that edges between them point mainly in one direction,and only a few point backwards, have a very low width.

iv) The last feature (iii) also made the algorithms to be XP, instead of FPT,because of the need to remember the partial results for all vertices withincoming edges from the outside, of which there could be |V |.

Directed tree-width. The first explicit directed measure was that of directed tree-width (dtw) [JRST01]. In the related cops-and-robber game the robber has tostay in the same cop-free strongly connected component, however the relation-ship between the number of cops needed and the directed tree-width is not strict.[JRST01] also contains XP algorithms for solving the Hamiltonian cycle, k-path,and related problems on graphs of bounded directed tree-width.

DAG-width. First defined in [Obd06] and, independently, in [BDHK06], DAG-width (dagw) was the next attempt to come up with a directed tree-width coun-terpart. This time the robber does not have to stay in the SCC, but the copstrategy has to be monotone, i.e., a cop cannot be placed on a previously va-cated vertex. This game fully characterizes DAG-width. Note that monotoneand non-monotone strategies are not equivalent [KO08].

Theorem 2.1 ([Obd06,BDHK06]). For any graph G, there is a DAG-decomposition of G of width k if, and only if, the cop player has a monotonewinning strategy in the k-cops-and-robber game on G.

Kelly-width. Defined a year later, Kelly-width (kellyw) [HK08] aimed to solve anexisting problem with DAG-decompositions: the number of nodes can be polyno-mially larger then the number of vertices in the original graph (the size dependson the width). The idea of Kelly-decompositions is based on the elimination or-dering for tree-width, and therefore the size of the decomposition is linear in thesize of the graph. The game characterizing Kelly-width is as for DAG-width, butwith two important differences: 1) the cops cannot see the robber, and 2) therobber can move only when a cop is about to land on his vertex.

Cycle rank. This is perhaps the oldest definition of a digraph connectivity mea-sure, given in 60’s by Eggan and Buchi [Egg63].

Definition 2.2 (Cycle rank). The cycle rank cr(G) of a digraph G is definedinductively as follows: For DAGs, cr(G) = 1. If G is strongly connected andE(G) 6= ∅, then cr(G) = 1 + min{ cr(G − v) : v ∈ V (G) }. Otherwise, cr(G) isthe maximum over the cycle rank of the strongly connected components of G.

Measure comparison. All the measures presented above are closely related toeach other. The following theorem in a summary shows that if a problem is hardfor graphs of bounded cycle rank, then it is hard for all the other measures.

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Theorem 2.3. Let G be a digraph. Then (dpw [Bar06] is the directed path-width):

1/3(dtw(G) − 1) ≤[BDHK06] dagw(G) ≤ dpw(G) ≤[Gru08] cr(G)

1/6(dtw(G) + 2) ≤[HK08] kellyw(G) ≤ dpw(G) ≤[Gru08] cr(G)

Moreover, when DAG-width is bounded, so is Kelly-width [HO06].

2.1 Directed rank-width

The rank-width of undirected graphs was introduced by Oum and Seymour inrelation to graph clique-width. While the definition of clique-width works “as is”also on digraphs, the following straightforward generalization of rank-width todigraphs (related to clique-width again) has been proposed by Kante [Kan08].

Definition 2.4 (Bi-rank-width). Consider a digraph G, and vertex subsetsX ⊆ V (G) and Y = V (G) \X. Let A+

X denote the X × Y 0, 1-matrix with theentries ai,j = 1 (i ∈ X, j ∈ Y ) iff (i, j) ∈ E(G), and let A−

X = (A+Y )T . The

bi-cutrank function of G is defined as the sum of the ranks of these two matricesbrkG(X) = rk(A+

X) + rk(A−X) over the binary field GF (2). The bi-rank-width

brwd(G) of G then equals the branch-width of this bi-cutrank function brkG.

We remind the readers that the branch-width [RS91] of an arbitrary symmet-ric submodular function λ : 2E → N is defined as the minimum width over allbranch-decompositions of λ over E, where a branch-decomposition is a pair T, τsatisfying the following: T is a tree of degree at most three, and τ is a bijectionfrom E to the leaves of T . If f is an edge of T , then let Xf ⊆ V (T ) be the vertexset of one of the two connected components of T − f , and let the width of f beλ(τ−1(Xf )). The width of T, τ is the largest width over all edges of T .

Importantly, as proved by Kante [Kan08], the rank-decomposition algorithmof [HO08] can also be used to find an optimal bi-rank-decomposition of a digraph.

Theorem 2.5 ([HO08] and [Kan08]). Let t ∈ N be constant. There existsan algorithm that in time O(n3), for a given n-vertex graph (digraph) G, eitheroutputs a rank-decomposition (bi-rank-decomposition, respectively) of G of widthat most t, or certifies that the rank-width (bi-rank-width) is more than t.

A rank-decomposition is, actually, not so suitable for designing dynamic pro-gramming algorithms. Yet, there is an efficient alternative characterization of arank-decomposition via algebraic terms (or parse trees) over the bilinear graphproduct, which has been proposed by Courcelle and Kante [CK07] and furtherextended towards algorithmic applications by [GH08] (see also an independentsimilar approach of [BXTV08]). As shown in [Kan08], an analogous “dynamicprogramming friendly” parse-tree view (of bi-rank-width) exists for digraphs,and we will apply this later, e.g. in Theorems 3.7 and 3.12.

2.2 DAG-depth

This part is inspired by the tree-depth notion of Nesetril and Ossona de Mendez.[NdM06, Lemma 2.2] gives an inductive definition of the tree-depth td(G) of

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undirected G as follows (compare to Def. 2.2). If G has one vertex, then td(G) =1. If G is connected, then td(G) = 1 + min{ td(G− v) : v ∈ V (G) }. Otherwise,td(G) equals the maximum over the tree-depth of the components of G.

We propose a new “directed” generalization of this definition. For a digraphG and any v ∈ V (G), let Gv denote the subdigraph of G induced by the verticesreachable from v. The maximal elements of the poset {Gv : v ∈ V (G) } in thegraph-inclusion order are called reachable fragments of G. Notice that reachablefragments in the undirected case coincide with connected components.

Definition 2.6 (DAG-depth). The DAG-depth ddp(G) of a digraph G is in-ductively defined: If |V (G)| = 1, then ddp(G) = 1. If G has a single reachablefragment, then ddp(G) = 1 + min{ddp(G− v) : v ∈ V (G) }. Otherwise, ddp(G)equals the maximum over the DAG-depth of the reachable fragments of G.

Comparing Definitions 2.2 and 2.6, one can see that DAG-depth equals cyclerank on bidirected orientations of graphs. Furthermore, the following useful gamecharacterization of this new measure can be proved along Definition 2.6.

Theorem 2.7. The DAG-depth of a digraph G is at most t if, and only if, thecop player has a “lift-free” winning strategy in the k-cops and robber game on G,i.e., a strategy that never moves a cop from a vertex once he has landed.

Corollary 2.8 (cf. Theorem 2.1, Def. 2.2). For any digraph G, the DAG-depth of G is greater than or equal to the DAG-width and the cycle rank of G. ⊓⊔

Another claim tightly relates our new measure to directed paths in a digraph.

Proposition 2.9. Consider a digraph G of DAG-depth t, and denote by ℓ thenumber of vertices of a longest directed path in G. Then ⌊log2 ℓ⌋ + 1 ≤ t ≤ ℓ.

2.3 K-width

Moreover, applications in various “directed path” problems, see e.g. Section 3.1,inspired the following width measure: The K-width (a shortcut of “Kennywidth”) of a digraph G is the maximum number of distinct (not necessarilydisjoint) simple s–t paths in G over all pairs of distinct vertices s, t ∈ V (G).

Similarly to DAG-depth in Proposition 2.9, K-width can be arbitrarily largeon DAGs. By giving a suitable search strategy for the cop player in a di-graph G based on a DFS tree of G, we show that K-width is lower-boundedby DAG-width, but K-width is generally incomparable with cycle-rank which isunbounded on bidirected paths.

Theorem 2.10 (cf. Theorem 2.1). For any digraph G, the K-width of G isgreater or equal to the DAG-width of G minus one.

Furthermore, an easy algorithm enumerating all paths leads to:

Proposition 2.11. The K-width k of a given digraph G can be computed intime k · poly(|V (G)|).

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3 Summary of Complexity Results

Table 1. Old and new (in boldface) complexity results on digraph measures ( ∗-markedresults assume a decomposition is given in advance; p-NPC is a shortcut for the com-plexity class para-NPC; and c and φ are fixed parameters of the respective problems).

Problem K-width DAG-depth DAG-width Cycle-rank DAG Bi-rank-width

HAM FPT FPT XPa ∗ XPa ∗ P XPb

W[2]-hardc W[2]-hardd

c-Path FPT FPT XPa ∗ XPa ∗ Pa FPT

k-Path p-NPC p-NPC NPC NPC NPC open

DiDS p-NPC p-NPC NPC NPC NPC FPT

DiSTP p-NPC p-NPC NPC NPC NPC FPT

MaxDiCut p-NPCc p-NPCc NPCc NPCc NPCc XP

c-OCN p-NPC p-NPC NPCe NPCe NPCe FPT

DFVS open open p-NPCf p-NPCf P FPT

Kernel p-NPCg p-NPCg p-NPCf ,g p-NPCf ,g P FPT

φ-MSO1mc p-NPH p-NPH NPH NPH NPH FPTh

φ-LTLmc p-coNPH p-coNPH coNPH coNPH coNPC p-coNPH

Parity XPi XPi XPi ∗ XPi ∗ P XPj

References a[JRST01] b[GH09] c[LKM08] d[FGLS09] e[CD06] f [KO08] g[vL76]h[CMR00] i[BDHK06] j[Obd07] . Refer to the respective following sections for detailsand the new results.

3.1 Hamiltonian Path (HAM) and Disjoint Paths (k-Path)

The classical NP-hard Hamiltonian Path (HAM) problem [GJ79] is to find adirected path that visits each vertex of a digraph exactly once. A natural gen-eralization of HAM is the Longest Path problem (Longest Path), where oneis asked to find the longest simple path in a given digraph.

It is easy to see that HAM can be solved on DAGs in polynomial time. Whenusing the parameter DAG-width, the problem belongs to the complexity classXP [JRST01], but was also proven to be W[2]-hard [LKM08]. We prove our newFPT results for the parameters K-width and DAG-depth on the more generalLongest Path problem. Using a simple enumeration of all distinct paths in thecase of bounded K-width, or applying Proposition 2.9 and any FPT-algorithm forLongest Path in the standard parameterization (e.g. [CKL+09]) when DAG-depth is bounded, we get:

Theorem 3.1. There is a fixed parameter tractable algorithm solving theLongest Path problem on a digraph G

a) in time O(

t · |V (G)| · |E(G)|)

if G is of K-width at most t;

b) in time O(

42t+O(t3) · |V (G)| · |E(G)|)

if G is of DAG-depth at most t.

Another well-known problem is Disjoint Paths (k-Path); given a digraphand k pairs of nodes (si, ti), 1 ≤ i ≤ k, the task is to find pairwise disjointdirected paths from each si to the respective ti. This problem is NP-complete

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[FHW80] even when k is bounded by any constant c ≥ 2 (c-Path). Moreover, a“mixed” generalization of c-Path remains NP-complete [BJK09] even on DAGs.

If the digraph of an instance of k-Path has K-width ≤ 2, then it can beexpressed as a 2-SAT formula, and if DAG-depth is ≤ 2, then it is equivalentto an SDR instance (system of distinct representatives). If, however, we slightlyrelax the restrictions as follows, the problem becomes NP-complete again.

Theorem 3.2. The k-Path problem (with k as part of input)a) can be solved in polynomial time on graphs of K-width or DAG-depth 2;b) is NP-complete on DAGs of K-width 3 and DAG-depth 4.

Finally, since one can express an instance of c-Path for any fixed c in MSO1

logic (Section 3.6), it follows from Theorem 3.12 that this problem is fixed pa-rameter tractable on digraphs of bi-rank-width t with parameters c and t. Thec-Path problem however also becomes easier for the other new measures:

Theorem 3.3. There is a fixed parameter tractable algorithm (for constant c)solving the c-Path problem on a digraph G

a) in time O(tc · |E(G)|) if G is of K-width at most t;

b) in time O(

(2c)ct4t

· |E(G)|2) if G is of DAG-depth at most t.

3.2 Directed Dominating Set (DiDS) and Steiner Tree (DiSTP)

The well-known NP-hard Dominating Set (DS) and Steiner Tree (STP) prob-lems both allow for natural directed counterparts. We consider them in their un-weighted variants for simplicity. The Directed Dominating Set problem (DiDS)asks for a minimum cardinality vertex set X in a digraph G such that everyvertex of G not in X is an outneighbour of X. The Directed Steiner Tree prob-lem (DiSTP) [HRW92], given a digraph G and T ⊆ V (G), r ∈ V (G), asks for aminimum size tree in G spanning r ∪ T with all arcs oriented away from r.

While it is folklore that both of these problems are NP-hard in general, weshow (with a simple reduction from Vertex Cover) that the same holds evenon very restricted graph classes.

Theorem 3.4. DiDS and DiSTP problems are NP-complete on a digraph Geven if G is restricted to be a DAG of K-width 2 and DAG-depth 3.

Applying the MSO1 optimization framework described in Section 3.6 we get:

Proposition 3.5 (Theorem 3.12). The (unit cost) DiDS and DiSTP prob-lems are fixed parameter tractable when parameterized by bi-rank-width.

3.3 Maximum directed cut (MaxDiCut)

Maximum directed cut (MaxDiCut) is an extensively studied problem on di-graphs. Given a digraph G, the goal is to partition the vertex set V (G) intoV0 and V1 such that the cardinality of { (u, v) ∈ E(G) : u ∈ V0, v ∈ V1 } is

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maximized. This problem is often stated with edge weights, but we consideronly the unweighted (cardinality MaxDiCut) variant in our paper.

It is well known that the MaxDiCut optimization problem is NP-hard, andit has been shown that MaxDiCut stays NP-hard even on DAGs [LKM08].A closer, yet quite nontrivial look, at the reduction reveals the resulting graphto have also bounded DAG-depth and K-width.

Theorem 3.6 ([LKM08]). The MaxDiCut problem is NP-hard on a digraphG even if G is restricted to be a DAG of K-width 4608 and DAG-depth 11.

The only new efficiently solvable case among our measures is the following:

Theorem 3.7. The unweighted MaxDiCut problem on a digraph G of bi-rank-width t is polynomially solvable for every fixed t (i.e. it belongs to the class XP).

3.4 Oriented Colouring (OCN)

A natural directed generalization of the ordinary graph colouring problem canbe obtained as follows: The chromatic number χ(G) of a graph G equals theminimum c such that G has a homomorphism into the complete graph Kc. TheOriented Chromatic Number (OCN) χo(G) of a digraph G is defined as theminimum c such that G has a homomorphism into some(!) orientation of Kc.

In other words, χo(G) equals minimum c such that the vertex set of G canbe partitioned into c independent sets such that, between each pair of the sets,all arcs have the same direction. For instance, χo = 5 for the directed 5-cycle.

It has been shown [KM04] that checking χo(G) ≤ 3 is easy, but determiningwhether χo(G) ≤ 4 is already NP-complete. Subsequently, [CD06] have shownthat the problem χo(G) ≤ 4 remains NP-complete even on acyclic digraphs.Using a simpler and more powerful reduction than [CD06], we prove:

Theorem 3.8. The problem (4-OCN) to decide whether a digraph G satisfiesχo(G) ≤ 4 is NP-complete even if G is a DAG of K-width 3 and DAG-depth 5.

On the other hand, it follows from the general framework of Theorem 3.12:

Proposition 3.9. The problem (c-OCN) to decide χo(G) ≤ c on an input di-graph G of bi-rank-width t is fixed parameter tractable with parameters c and t.

3.5 Directed Feedback Vertex Set (DFVS) and Kernel (Kernel)

The directed feedback vertex set (DFVS) problem is to find a minimum cardina-lity set S of vertices of a digraph G whose removal leaves G \ S acyclic. Thisproblem is trivial for acyclic digraphs, and it is FPT with the parameter k = |S|.We hence consider only the optimization variant of DFVS with unbounded k.

Kreutzer and Ordyniak [KO08] gave a reduction showing NP-hardness of theDFVS optimization problem on digraphs of DAG-width 4. A closer look at thisreduction reveals that all the produced graphs are moreover of cycle rank 4, butthey have unbounded K-width and DAG-depth.

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The kernel of a digraph G is defined as an independent set S ⊆ V (G) suchthat for every x ∈ V (G) \ S there is an arc from x into S. Notice that a kernelmay not always exist. However, on acyclic digraphs, a kernel can be easily found.Having a closer look at the NP-completeness reduction of van Leeuwen [vL76],one discovers the following claim (cf. also [KO08]).

Theorem 3.10 (van Leeuwen [vL76]). It is NP-complete to decide whethera digraph G has a kernel, even if G is restricted to have (all at once) DAG-widthand K-width 2, cycle rank also 2, and DAG-depth 4.

Finally, by Example 3.11 and Theorem 3.12, both the Kernel and DFVS

problems are fixed parameter tractable on digraphs of bounded bi-rank-width.

3.6 MSO1 Model Checking (φ-MSO1mc)

Monadic second order (MSO) logic is a language often used for description ofcombinatorial algorithmic problems. When applied to a one-sorted relationalgraph structure (i.e. to a set V with a symmetric relation edge(u, v)), this lan-guage is abbreviated as MSO1. We use the same abbreviation MSO1 also fordigraphs with a relation arc(u, v).

Example 3.11. The following properties are expressible in MSO1 on digraphs

– a directed dominating set X as ∀z(

z ∈ X ∨ ∃x ∈ X arc(x, z))

,

– the existence of a kernel S as ∃S ∀x[

x 6∈ S ↔(

∃y ∈ S arc(x, y))]

, or

– a feedback vertex set Z as ∀X[

X ∩ Z = ∅ →(

∃x ∈ X ∀y ∈ X ¬arc(x, y))]

.

On the other hand, MSO1 cannot express Hamiltonian cycle, for instance.

The MSO1 model checking problem (φ-MSO1mc), where φ is a fixed for-mula, is FPT on (undirected) graphs of bounded clique-width or rank-width[CMR00,CK07]. Not surprisingly, this extends to digraphs parameterized by bi-rank-width. More generally, the LinEMSO1 optimization framework includes allproblems which can be expressed as maximization of a linear evaluational termover all tuples of sets X1, . . . ,Xj satisfying ψ(X1, . . . ,Xj) where ψ is an MSO1

formula —see [CMR00] for details. Analogously to [CMR00] (or [GH08]) we get:

Theorem 3.12 (cf. [CMR00], and [Kan08,GH08]).Every ψ-LinEMSO1 optimization problem is fixed parameter tractable when re-stricted to digraphs of bi-rank-width t, with parameters t and ψ.

Theorem 3.12 particularly implies that the problems listed in Example 3.11(and many others) are FPT on digraphs of bi-rank-width t. No analogous results,however, seem possible for our other directed width measures since one caninterpret φ-MSO1mc of arbitrary undirected graphs via subdividing each edgeand giving the two new edges opposite orientations, leading to:

Proposition 3.13. The φ-MSO1mc problem is NP-hard even when restrictedto DAGs that are of K-width 1 and DAG-depth 2.

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3.7 LTL Model Checking (φ-LTLmc) and Parity Games (Parity)

Another useful language that allows to express properties of digraphs is LinearTemporal Logic (LTL) —see, e.g., [BK08]. LTL model checking remains hard fora fixed formula φ and all of the directed width measures we considered here,including bi-rank-width (as opposed to MSO1 model checking).

Theorem 3.14. The φ-LTLmc problem is coNP-hard even when the input di-graph is restricted to have K-width 1, DAG-depth 4, and bi-rank-width 2.

Theorem 3.15. The φ-LTLmc problem is coNP-complete on DAGs.

Parity games—see e.g. [GTW02] for a reference, play an important role inthe field of model-checking and formal verification. There are many reasons forthis. First, solving parity games is equivalent to model-checking the modal µ-calculus, an important modal logic subsuming many other logics (e.g. CTL).Moreover, the modal µ-calculus is a bisimulation invariant fragment of MSO1.

Second, the exact complexity of solving a parity game is a long-standing openproblem. It is known to be in NP∩ co-NP, and widely believed to be in P . It istrivially in P for acyclic digraphs. Moreover, it was shown that solving a paritygame is in XP for digraphs of bounded tree-width [Obd03], bounded DAG-width[BDHK06] (hence also on bounded K-width, DAG-depth, and cycle rank) andbounded Kelly-width [HK08], and of bounded clique-width [Obd07] (implyingthe same for bi-rank-width).

4 Conclusion

Table 1, and the related results in this paper, have left several interesting openproblems and questions. Just to specifically mention a few:

1) We suggest there exist FPT algorithms solving the DFVS problem forbounded K-width or DAG-depth (two of the open table entries).

2) For some entries in the table, we neither expect an FPT algorithm, nor havean NP-hardness estimate. E.g., MaxDiCut or k-Path for bi-rank-width, orc-Path for cycle rank. Can we then, at least, show a W-hardness result?

3) While we have given FPT and XP, respectively, algoritms solving the unit-cost variants of DiSTP and MaxDiCut, these problems are usually consid-ered in their weighted variants and then we expect their complexity to behigher. We, however, have no further results in this direction.

4) Some suggest that the DFVS number (see in Section 3.5) perhaps can be agood directed width measure. However, since majority of our sample prob-lems in Table 1 remain hard even on DAGs, there is not much room left forapplications of the DFVS parameter. Interestingly though, Kernel becomesFPT when parametrized by DFVS.

Theorem 4.1. If a digraph G is given with a directed feedback vertex set ofsize k, then the Kernel problem can be solved in time O(2k · |V (G)|2).

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Finally, we try to formulate the overall impression coming from Table 1:Robber-and-cops based width measures do not seem to be very useful for pa-rameterized algorithms on digraphs. One reason might be that cops “give” goodgraph separators in the undirected case, but that does not work any more fordigraphs. Considering the DFVS number as a width parameter does not seemto help either. We perhaps need something new to move on. At this moment,bi-rank-width seems like a good alternative.

Acknowledgements

This work has been supported by a Czech–German bilateral grant of GACRand DFG (201/09/J021 and RO 927/9). Moreover, P. Hlineny has also beensupported by the Czech research grant GACR 201/08/0308.

References

[Bar06] J. Barat. Directed path-width and monotonicity in digraph searching.Graphs and Combinatorics, 22(2):161–172, 2006.

[BDHK06] D. Berwanger, A. Dawar, P. Hunter, and S. Kreutzer. DAG-width andparity games. In STACS’06, volume 3884 of LNCS, pages 524–536. Springer,2006.

[BG04] D. Berwanger and E. Gradel. Entanglement – a measure for the complexityof directed graphs with applications to logic and games. In LPAR 2004,volume 3452 of LNCS, pages 209–223. Springer, 2004.

[BJK09] J. Bang-Jensen and M. Kriesell. Disjoint directed and undirected pathsand cycles in digraphs. Technical Report PP-2009-03, University of SouthDenmark, 2009.

[BK08] C. Baier and J.-P. Katoen. Principles of Model Checking. The MIT Press,Cambridge, Massachusetts, 2008.

[BXTV08] B.-M. Bui-Xuan, J. Telle, and M. Vatshelle. H-join and algorithms ongraphs of bounded rank-width. submitted, November 2008.

[CD06] J.-F. Culus and M. Demange. Oriented coloring: Complexity and approx-imation. In SOFSEM’06, volume 3831 of LNCS, pages 226–236. Springer,2006.

[CK07] B. Courcelle and M. Kante. Graph operations characterizing rank-widthand balanced graph expressions. In WG’07, volume 4769 of LNCS, pages66–75. Springer, 2007.

[CKL+09] J. Chen, J. Kneis, S. Lu, D. Molle, S. Richter, P. Rossmanith, S. Sze, andF. Zhang. Randomized divide-and-conquer: Improved path, matching, andpacking algorithms. SIAM Journal on Computing, 38(6):2526–2547, 2009.

[CMR00] B. Courcelle, J. A. Makowsky, and U. Rotics. Linear time solvable optimiza-tion problems on graphs of bounded clique-width. Theory Comput. Syst.,33(2):125–150, 2000.

[DGK08] P. Dankelmann, G. Gutin, and E. Kim. On complexity of minimun leafout-branching. arXiv:0808.0980v1, August 2008.

[Egg63] L. Eggan. Transition graphs and the star-height of regular events. Michigan

Mathematical Journal, 10(4):385–397, 1963.[FGLS09] F. Fomin, P. Golovach, D. Lokshtanov, and S. Saurab. Clique-width: On

the price of generality. In SODA’09, pages 825–834. SIAM, 2009.

11

[FHW80] S. Fortune, J. E. Hopcroft, and J. Wyllie. The directed subgraph homeo-morphism problem. Theor. Comput. Sci., 10:111–121, 1980.

[GH08] R. Ganian and P. Hlineny. Automata approach to graphs of bounded rank-width. In IWOCA’08, pages 4–15, 2008.

[GH09] R. Ganian and P. Hlineny. Better polynomial algorithms on graphs ofbounded rank-width. In IWOCA’09, LNCS. Springer, 2009. to appear.

[GJ79] M. Garey and D. Johnson. Computers and Intractability: A Guide to the

Theory of NP-completeness. Freeman, San Francisco, 1979.[Gru08] H. Gruber. Digraph complexity measures and applications in formal lan-

guage theory. In MEMICS’08, pages 60–67, 2008.[GTW02] E. Gradel, W. Thomas, and T. Wilke, editors. Automata, Logics, and Infi-

nite Games, volume 2500 of LNCS. Springer, 2002.[HK08] P. Hunter and S. Kreutzer. Digraph measures: Kelly decompositions, games,

and orderings. Theor. Comput. Sci., 399(3):206–219, 2008.[HO06] P. Hlineny and J. Obdrzalek. Escape-width: Measuring ”width” of digraphs.

Presented at Sixth Czech-Slovak International Symposium on Combina-torics, Graph Theory, Algorithms and Applications, 2006.

[HO08] P. Hlineny and S. Oum. Finding branch-decomposition and rank-decomposition. SIAM J. Comput., 38:1012–1032, 2008.

[HRW92] F. Hwang, D. Richards, and P. Winter. The Steiner Tree Problem. Annalsof Discrete Mathematics. Noth-Holland, 1992.

[JRST01] T. Johnson, N. Robertson, P. D. Seymour, and R. Thomas. Directed tree-width. Journal of Combinatorial Theory, Series B, 82(1):138–154, 2001.

[Kan08] M. Kante. The rank-width of directed graphs. arXiv:0709.1433v3, March2008.

[KM04] W. Klostermeyer and G. MacGillivray. Homomorphisms and oriented col-orings of equivalence classes of oriented graphs. Discrete Mathematics,274:161–172, 2004.

[KO08] S. Kreutzer and S. Ordyniak. Digraph decompositions and monotonicityin digraph searching. In WG’08, volume 5344 of LNCS, pages 336–347.Springer, 2008.

[LKM08] M. Lampis, G. Kaouri, and V. Mitsou. On the algorithmic effectiveness ofdigraph decompositions and complexity measures. In ISAAC, volume 5369of LNCS, pages 220–231. Springer, 2008.

[NdM06] J. Nesetril and P. Ossona de Mendez. Tree-depth, subgraph coloring andhomomorphism bounds. European J. Combin., 27(6):1024–1041, 2006.

[Obd03] J. Obdrzalek. Fast mu-calculus model checking when tree-width is bounded.In CAV 2003, volume 2725 of LNCS, pages 80–92. Springer, 2003.

[Obd06] J. Obdrzalek. DAG-width – connectivity measure for directed graphs. InSODA’06, pages 814–821. ACM-SIAM, 2006.

[Obd07] Jan Obdrzalek. Clique-width and parity games. In CSL’07, volume 4646 ofLNCS, pages 54–68. Springer, 2007.

[RS86] N. Robertson and P. D. Seymour. Graph minors. II. Algorithmic aspects oftree-width. Journal of Algorithms, 7(3):309–322, September 1986.

[RS91] N. Robertson and P. D. Seymour. Graph minors. X. Obstructions to tree-decomposition. J. Comb. Theory B, 52(2):153–190, 1991.

[Saf05] M. Safari. D-width: A more natural measure for directed tree-width. InMFCS’05, volume 3618 of LNCS, pages 745–756. Springer, 2005.

[vL76] J. van Leeuwen. Having a Grundy-numbering is NP-complete. TechnicalReport 207, The Pennsylvania State University, September 1976.

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APPENDIX

Notes on Section 2.1 (bi-rank-width)

A rank-decomposition is, actually, not so suitable for designing dynamic pro-gramming algorithms. Yet, there is an efficient alternative characterization of arank-decomposition via algebraic terms (or parse trees) over the bilinear graphproduct, which has been proposed by Courcelle and Kante [CK07] and furtherextended towards algorithmic applications by [GH08] (see also an independentsimilar approach of [BXTV08]). As shown in [Kan08], an analogous “dynamicprogramming friendly” parse-tree view (of bi-rank-width) exists for digraphs,and we will apply this later, e.g. in Theorem 3.12. in Section 3.6.

Following Section 2.1, we describe Kante’s bi-labeling parse trees [Kan08,Section 4] (thereafter called “algebraic expressions for bin-rank-width”),which characterize bi-rank-width of digraphs up to a multiplicative factor 2(Lemma 5.3).

A t-labeled digraph is a pair G = (G, lab) of a digraph G and a vertex labelinglab : V (G) → 21,...,t into subsets of t labels, or equivalently in linear algebraterms a mapping lab : V (G) → GF (2)t into the points of a t-dimensional binaryvector space. For technical reasons, we analogously define a t-bi-labeled digraphH = (H, lab+, lab−). A t-relabeling is a linear mapping f : GF (2)t → GF (2)t, orin other words a binary t× t matrix f .

Definition 5.1 (Bi-labeling join). Considering a t-labeled digraph G =(G, lab) and a t-bi-labeled digraph H = (H, lab+, lab−), a t-bi-labeling join G⊗His defined on a disjoint union of G and H by adding, where u ∈ V (G), v ∈ V (H);all arcs (u, v) such that |lab(u) ∩ lab+(v)| is odd, and all arcs (v, u) such that|lab(u) ∩ lab−(v)| is odd. The resulting digraph is unlabeled.

Definition 5.2 (Bi-labeling parse trees). Considering t-labeled digraphsGi = (Gi, labi), i = 1, 2, and relabelings f1, f2, h

+, h− : GF (2)t → GF (2)t,we define a t-bi-labeling composition operator ⊗[h+, h−; f1, f2] as follows.G1 ⊗ [h+, h−; f1, f2] G2 = G3 where G3 = G1 ⊗ (G2, h

+· labT2 , h−· labT2 ) and the

labeling of v ∈ V (Gi) in G3 is lab3(v) = fi · labTi , i = 1, 2.

A t-bi-labeling parse tree T , see also [GH08], is a finite rooted ordered sub-cubic tree (with the root degree at most 2) such that

– the leaves of T contain a ⊙ symbol creating a new graph vertex of label {1},– each internal node of T contains one of the t-bi-labeling composition symbols.

A parse tree T then generates (parses) the digraph G which is obtained bysuccessive leaves-to-root applications of the operators in the nodes of T .

Lemma 5.3 (Kante [Kan08]). Let G be a digraph of bi-rank-width t. If m isthe smallest integer such that (some labeling of) G is produced by some m-bi-labeling parse tree, then t ≥ m ≥ t/2.

13

Having now the bi-labeling parse tree machinery at hand, it is straightforwardto translate the formal tools of [GH08,GH09] to digraphs of bounded bi-rank-width, see e.g. the proof of Theorem 3.7. In this way, for instance, the XPalgorithm for undirected Hamiltonian path [GH09] directly translates to an XPalgorithm for Hamiltonian path in digraphs of bounded bi-rank-width.

On the other hand, we remind the readers that one cannot use an undi-rected rank-decomposition (or parse tree) of a digraph G to design a dynamicprogramming algorithm for a problem referring to the direction of arcs of G.That is because the parse tree produces large bipartite cliques, and one cannotexhaustively process all possible orientations of those.

For sake of completeness, we lastly remark that Kante [Kan08] considers alsoanother directed generalization of rank-width, the so called GF (4)-rank-width.Since these two are within a constant factor, there is no need to consider thelatter in our paper.

Proofs for Section 2.2 (DAG-depth)

Theorem 2.7. The DAG-depth of a digraph G is at most t if, and only if, thecop player has a “lift-free” winning strategy in the k-cops and robber game on G,i.e., a strategy that never moves a cop from a vertex once he has landed.

Proof. (sketch) We proceed by induction along the definition of DAG-depth.That is trivial if |V (G)| = 1. Let F1, . . . , Fd be all the reachable fragments of G. Ifd > 1, then the robber may start in any vertex of any Fi ⊆ G, i ∈ {1, . . . , d}, andso the cop player needs as many moves in G as in such most expensive reachablefragment which is max{ddp(Fi) : i = 1, . . . , d } by inductive assumption.

Now assume G has a single reachable fragment. Hence there is v ∈ V (G)which can reach whole G, and so whenever another cop is to land at s ∈ V (G),the robber may move to any vertex of G−s. It follows from inductive assumptionthat the cop player needs another ddp(G−s) moves after landing at s. Therefore,the cop player needs at least 1 + min{ddp(G− v) : v ∈ V (G) } moves on G, andthis is also sufficient. ⊓⊔

Proposition 2.9. Consider a digraph G of DAG-depth t, and denote by ℓ thenumber of vertices of a longest directed path in G. Then ⌊log2 ℓ⌋ + 1 ≤ t ≤ ℓ.

Proof. Firstly, we show that the DAG-depth of an ℓ-vertex path P is at least⌊log2 ℓ⌋+1. This is trivial if ℓ = 1. Since a path has a single reachable fragment,we have from the definition ddp(P ) = 1 + ddp(Q) where Q a path of length⌈(ℓ− 1)/2⌉. If ℓ is even, then ddp(P ) = 1 + ⌊log2(ℓ/2)⌋ + 1 = ⌊log2 ℓ⌋+ 1. If ℓ isodd, then ddp(P ) = 1 + ⌊log2((ℓ− 1)/2)⌋ + 1 = ⌊log2(ℓ− 1)⌋ + 1 = ⌊log2 ℓ⌋+ 1.

On the other hand, we describe a simple ℓ-move lift-free winning strategy forthe cop player on any such digraph G. The first cop lands on the initial positions1 of the robber. In cop move i > 1, the cop number i lands on a vertex si ofG which is the out-neighbour of si−1 on some directed path from si−1 to thecurrent robber position. Since all directed paths starting in s1 have ≤ ℓ vertices,the robber is finally caught after ≤ ℓ cop moves. ⊓⊔

14

Notice that Proposition 2.9 provides efficient computation of DAG-depth:

Corollary 5.6. The DAG-depth of a digraph G can be approximated by an FPTalgorithm, and computed exactly by an XP algorithm.

Proof. We first compute the longest directed path length ℓ in G, which can bedone by an FPT algorithm of e.g., [CKL+09]. This ℓ is already a good estimateof the DAG-depth of G by Proposition 2.9.

In the second part, we carry a brute-force recursive computation of the DAG-depth of G according to the definition. Since the depth of recursion is boundedby ℓ, and each call branches into O(|V (G)|) subproblems, we get an XP algo-rithm. ⊓⊔

Furthermore, notice that the approach of Corollary 5.6 also gives an efficientway to construct a bounded DAG-decomposition for G if the DAG-depth isbounded (of course, with no matching lower bound).

Proofs for Section 2.3 (K-width)

Theorem 2.10. For any digraph G, the K-width of G is greater or equal to theDAG-width of G minus one.

Proof. Let T be any depth first search tree of G. Based on T , we outline amonotone search strategy for the cop player on G, in which the player is to usea cop number k + 1 only if there are at least k paths between a pair of vertices.

(i) In the first move a cop is placed at the root of T .(ii) In each subsequent cop-placing move, the cop player chooses the (unique)

vertex v of G such that; v is an outneighbour of a cop-occupied vertex, andv reaches the robber along a cop-free path in T .

(iii) Whenever a cop-occupied vertex u is no longer reachable from the currentrobber position, the cop from u is lifted back.

This strategy is clearly monotone. Consider the vertex v in rule (ii). If therewas a cop-occupied vertex w in G which is not an ancestor of v, then w mustno longer be reachable by the robber since T is a DFS tree. So (iii) for u = wapplies before (ii). Therefore, our strategy maintains an invariant that all verticesoccupied by cops belong to one directed path of T .

Consider now a situation when there is a set U of k cop-occupied vertices in G,and rule (ii) applies again. Then there is a path P ⊆ T such that U ⊆ V (P ). Lets be the last cop-occupied vertex of P . By (iii), each vertex w ∈ U is reachablein G from the robber vertex r along a cop-free path Qw. So P ∪Qw contains apath from r to s, and these k paths are pairwise distinct for distinct w. ⊓⊔

Notice that the proof of Theorem 2.10, together with Proposition 2.11, givean efficient way to construct a bounded DAG-decomposition for G if the K-widthis bounded. Furthermore, the following simple claim will be useful in algorithmicapplications of K-width.

15

Lemma 5.8. If G is a digraph of K-width t, then all (at most t|V (G)|) directedpaths starting at a vertex u ∈ V (G) can be enumerated in time O(t · |E(G)|).

Proof. Enumerate paths in G starting at u by backtracking and prune the searchwhenever finding a vertex that is already on the current path.

The resulting search tree has at most t|V (G)| nodes: Each node in the searchtree corresponds to a simple path in G starting at u. There can be at most tsuch paths with the same terminal vertex.

The time spent in each node of the search tree is O(d), where d is the out-degree of the terminal vertex of the corresponding simple path. Overall thisamounts to a running time of O(t|E(G)|). ⊓⊔

Finally, we can easily compute the K-width, if it is not too big.

Proposition 2.11. The K-width k of a given digraph G can be computed intime k · poly(|V (G)|).

Proof. Enumerate all (up to k each) simple paths starting from every vertexin G. Count how many path to each other vertex. The maximum number youencounter is exactly k. ⊓⊔

Proofs for Section 3.1 (HAM, k-Path)

Note that FPT-membership for Longest Path implies membership for HAM.

Theorem 3.1(a). Given a digraph G with K-width at most t, one can solveLongest Path in time O(t|V (G)| · |E(G)|).

Proof. For all u ∈ V (G) enumerate all simple paths starting at u according toLemma 5.8 while keeping track of their lengths. ⊓⊔

Theorem 3.1(b). Given a digraph G with DAG-depth at most t, one can solve

Longest Path in time 42t+O(t3) · |V (G)| · |E(G)|.

Proof. We know by Proposition 2.9 that ⌊log2 ℓ⌋+ 1 ≤ t ≤ ℓ, or in other words,ℓ ≤ 2t, where ℓ is the length of the longest path. We can hence use an arbitraryFPT-algorithm for the Longest Path decision problem in the standard param-eterization (e.g., [CKL+09] with running time 4ℓ+O(log3 ℓ)|V (G)| · |E(G)|): Webegin with ℓ = 1 and subsequently increase ℓ until a “no”-instance is found.This yields an FPT-algorithm for parameter t even if t is unknown to the algo-rithm. ⊓⊔

Theorem 3.2(a). The k-Path problem can be solved in polynomial time ongraphs of K-width at most 2 or DAG-depth at most 2.

Proof. Given a digraph G with K-width ≤ 2 and k pairs of nodes (s1, t1),. . . , (sk, tk), we first for every 1 ≤ i ≤ k compute by Lemma 5.8 the (wlog) twopossible paths pi,1 and pi,2 from si to ti. Then we construct a 2-SAT formula

16

sxi txi

xi,1 xi,2

xi,1 xi,2

· · ·

· · ·

......

sC1

tC1

sC2

tC2

sC3

tC3

sx0 tx0

sx1 tx1

sx2 tx2

Fig. 1. Left: gadget for variable xi; right: schematic of the construction

as follows: For each pair (si, ti), 1 ≤ i ≤ k, there is a clause over the two alter-native paths, Ci = {pi,1, pi2}. Furthermore, for each pair of non-disjoint pathsp1, p2 ∈ { pi,j : 1 ≤ i ≤ k, 1 ≤ j ≤ 2 }, such that p1 6= p2 and V (p1) ∩ V (p2) 6= ∅,there is a clause excluding each other, Cp1,p2

= {¬p1,¬p2}. We omit the sim-ple proof that the formula is satisfiable if and only if there is a solution to thek-paths instance at hand.

Similarly, given a digraph G with ddp(G) ≤ 2 and k pairs of nodes (s1, t1),. . . , (sk, tk), we proceed as follows. In the first step, for each pair (si, ti) suchthat (si, ti) ∈ E(G), we simple remove both vertices si, ti from the instance.

Hence we may assume that every si–ti path in G is formed by a pair of arcs(si, x), (x, ti) ∈ E(G), cf. Proposition 2.9. We denote by Xi the set of all suchx in G for the pair (si, ti). Then, clearly, the k-Path instance has a solution ifand only if X1, . . . ,Xk admit a system of distinct representatives, which can bedecided in P. ⊓⊔

Theorem 3.2(b). The k-Path problem is NP-complete on DAGs with K-width 3 and DAG-depth 4.

Proof. We reduce from the well-known NP-complete 3-SAT problem, where eachclause contains exactly three literals. Let ϕ be a 3-SAT-formula with m clausesC1, . . . , Cm over n variables. Without loss of generality, we may assume thatevery variable occurs in at most three literals (cf. the proof of Theorem 3.6),and that no variable has all three literals positive or all three negated (otherwisewe set it true or false, respectively). Hence every literal occurs at most two timesin the whole formula ϕ. We create a digraph G as follows.

For every variable xi, we add a gadget as depicted in Figure 1. The “up-per” path in the gadget corresponds to a negative assignment of the vari-able since it leaves the nodes xi,1 and xi,2 available for clauses, while simi-larly the “lower” path corresponds to a positive assignment. Then, for everyclause Ci = {ℓ1, ℓ2, ℓ3}, we add two nodes sCi

and tCi. Then, for every literal lj ,

17

1 ≤ j ≤ 3, such that lj is the kth occurrence in the formula, we add the edges(sCi

, ℓj,k) and (ℓj,k, tCi).

For example, if l3 = x5, and x5 occurred already in some Ci′ with i′ < i, thenwe add the edges (sCi

, x5,2) and (x5,2, tCi). See Figure 1 for a schematic view.

It is easy to see that the resulting digraph is a DAG, the longest path con-tains at most four nodes (which bounds the DAG-depth by Proposition 2.9),and between any two nodes there are at most three paths. Furthermore, ϕ issatisfiable if and only if G is a “yes”-instance to the k-path problem with pairs(sxi

, txi) for all 1 ≤ i ≤ n and pairs (sCj

, tCj) for all 1 ≤ j ≤ m:

Let C be a satisfying assignment of the variables. For the path between apair (sxi

, txi), we use the path corresponding to the assignment of the variable

xi, i.e., if xi is assigned 0, we use the path through the nodes labeled with xi,1

and xi,2, and the path through xi,1 and xi,2 otherwise. If a clause Cj is satisfiedby some literal li, then by construction the path between sxi

and txiis not using

the node v labeled with li, which means we can use the path sCjvtCj

for thepair (sCj

, tCj). Hence, all pairs can be connected by disjoint paths.If otherwise there is a solution to the k-path problem on the constructed

instance, then first note that a path between each sxiand txi

for every variable xi

either has to use the “positive” or the “negative” path through its correspondinggadget. We choose an assignment C of the variables, where each variable isassigned 0 if the path between sxi

and txiuses the path through the nodes

labeled with xi,1 and xi,2, and is assigned 1 else. Then each clause Cj = {l1, l2, l3}is satisfied: The path between sCj

and tCjhas to use one of the three nodes

corresponding to l1, l2, and l3, say lk for some variable xi. Since all paths aredisjoint, the path between sxi

and txiis not using lk, and therefore the variable

is assigned a value such that lk has the value 1 and Cj is satisfied. ⊓⊔

Lemma 5.14. Let G be a digraph, and let c pairs of vertices si, ti ∈ V (G), i =1, . . . , c be given. There is an MSO1 formula expressing (c-Path) the existenceof c pairwise disjoint directed si– ti paths, i = 1, . . . , c, in G.

Proof. We write

∃X1, . . . ,Xc

∧

i 6=j∈{1,...,c}

Xi ∩Xj = ∅ ∧∧

i∈{1,...,c}

si, ti ∈ Xi ∧

∧

i∈{1,...,c}

∀Z ⊆ Xi

(

(si ∈ Z ∧ ti 6∈ Z) → ∃x ∈ Z, y ∈ Xi\Z arc(x, y))

which means that there exist pairwise disjoint sets X1, . . . ,Xc ⊆ V (G) suchthat si, ti ∈ Xi, and each Xi induces a subdigraph of G in which ti is reachablefrom si. Notice that si, ti are not variables, but constants in this sentence. ⊓⊔

Theorem 3.3. There is a fixed parameter tractable algorithm solving thec-Path problem (for fixed c) on a digraph G

a) in time O(tc · |E(G)|) if G is of K-width at most t;

b) in time O(

(2c)ct4t

· |E(G)|2) if G is of DAG-depth at most t.

18

Proof. (sketch) Notice that we can, without loss of generality of the c-Path

problem, assume that G is a simple digraph (while 2-cycles are permitted).a) We, for each i = 1, . . . , c, use Lemma 5.8 to list all ≤ t distinct directed

paths from si to ti. Then, using brute force over all tc possibilities, we checkwhether there is a selection of pairwise disjoint ones.

b) This algorithm uses part (a) and recursive calls in a clever way. By Propo-sition 2.9, the longest directed path in G has length ℓ < 2t (which is the onlyextra information we use about G). We are actually going to recursively solvea more general problem to find a collection of c pairwise disjoint directed si– tipaths Qi in G such that E(Qi) ⊆ Ei ⊆ E(G). Initially E1 = · · · = Ec = E(G).

Let Pi be the collection of all si– ti paths with arcs from Ei. If |Pi| ≤ (cℓ)ℓ2

for all i = 1, . . . , c, then we may actually use (a) to solve the problem in time

O(

(cℓ)cℓ2 · |E(G)|)

. Otherwise, |Pi| > (cℓ)ℓ2 for some i, and we may apply thefollowing for u = si, v = ti, and (loosely) m = cℓ, k = ℓ:

Claim. Let H be a simple digraph, and u, v two vertices of H such that thelongest path starting in u has length k+ 1 and there exist 1+ (m− 1)k2

distinctdirected u–v paths in H. Then there exist vertices u′, v′ in H such that thereare m pairwise internally disjoint u′–v′ paths.

To prove the claim, we may assume that every arc of H is on some u–vpath. By the pigeon-hole priciple, there exists a vertex u′ in H having outdegree≥ 1+(m−1)k, and this u′ is not an in-neighbour of v (otherwise, we would have

only ≤(

(m−1)k)k

distinct u–v paths). Let u′i, i = 1, . . . , p ≥ 1+(m−1)k be theout-neighbours of u′ in H, and let H ′ be an inclusion-wise minimal subgraph ofH such thatH ′ contains some u′–v path Si passing through u′i for all i = 1, . . . , p.By a symmetric application of the pigeon-hole priciple, there exists a vertex v′

having indegree ≥ m in H ′. Let v′j , j = 1, . . . , q ≥ m be the in-neighbours of v′

in H ′. It follows from minimality of H ′ that the u′–v′ paths S′j passing through

appropriate u′ijand v′j are pairwise internally disjoint.

In other words, there exist vertices s′, t′ in G such that cℓ suitable fragmentsof paths from Pi form pairwise internally disjoint s′– t′ paths R1, . . . , Rcℓ. Thesepaths can be found in time O

(

(cℓ)ℓ2 · |E(G)|)

using an approach similar toLemma 5.8. Now, we make a new arc set E′

i from Ei by removing all arcs ofR1 ∪ · · · ∪ Rcℓ, and adding a new arc f ′ = (s′, t′). We call the same algorithmrecursively on E1, . . . , E

′i, . . . , Ec.

This algorithm clearly stops after O(c|E(G)|) recursive calls since each call

decreases |E1|+ · · · + |Ec|. Hence the overall run-time is O(

(cℓ)cℓ2 · |E(G)|2)

. Itremains to prove that there is a solution with constrains to E1, . . . , Ei, . . . , Ec

if, and only if, there is a solution to E1, . . . , E′i, . . . , Ec. The “only if” direction

is trivial since we can simply use the arc f ′ = (s′, t′) when needed.In the “if” direction, when f ′ is not used in the path, we are done. If f ′

is used in the si– ti path Q′i, then we notice the following: By the pigeon-hole

principle, some of the paths R1, . . . , Rcℓ must be disjoint from all other c − 1paths of ≤ ℓ vertices in the solution, and hence we can use the path (Qi−f

′)∪Rj

with all arcs in Ei instead. ⊓⊔

19

Proofs for Section 3.2 (DiDS and DiSTP)

Theorem 3.4. DiDS and DiSTP are NP-complete on DAGs that are of K-width 2 and DAG-depth 3.

Proof. We use a reduction from Vertex Cover to show hardness. Let a graphG = (V,E) and k ∈ N be an input instance for Vertex Cover. Wlog, we canassume that |V | ≥ k + 2.

We construct G′ = (V ′, E′) as follows. We set V ′ = V ∪E ∪ {v0} and definethe set of edges as follows:

E′ = { (v0, v) : v ∈ V } ∪ { (v, e) ∈ V × E : v ∈ e }.

Now, G = (V,E) has a vertex cover of size k iff G′ = (V ′, E′) has a directeddominating set of size k + 1.

Assume that there is some k vertex cover C ⊆ V in G. Then v0 ∪ C is adirected dominating set in G′, because v0 dominates itself as well as all v ∈ V ,and since each e ∈ E is incident to some v ∈ C, C dominates E in G′.

Now let D be a directed dominating set in G′ of size k+1. Since |V | ≥ k+2,v0 ∈ D, because otherwise a node in V would not be dominated. Moreover, wecan assume that D ∩E = ∅, because each e ∈ E can only dominate itself in G′.It is thus always safe to pick a predecessor of e instead. But then, each e ∈ E isdominated by some v ∈ D ∩ V , and thus D ∩ V is a vertex cover in G.

Finally, G′ is a DAG with K-width two, since there are only two paths fromv0 to each e ∈ E, only one path from v0 to each v ∈ V and only one path fromeach v ∈ V to each e ∈ E. Likewise, the DAG-depth of G′ is at most three.

Note that the same construction also can be used to prove hardness for theDiSTP problem. The dominating set implied by a vertex cover forms a Steinertree of size 1 + k + n, by connecting all e ∈ E via nodes in D to the root v0.

Moreover, any Steiner tree T that connects v0 to all e ∈ E implies a vertexcover V (T ) ∩ V , since each node in E must be connected by a node v ∈ V withv ∈ e. Moreover, any such Steiner tree T of cost at most k+ |E| contains at mostk nodes from V , and thus V (T ) ∩ V is a vertex cover of size k in G. ⊓⊔

Proposition 3.5. The (unit cost) DiSTP problem can be formulated as aLinEMSO1 optimization problem, and hence DiSTP is fixed parameter tractablewhen parameterized by bi-rank-width.

Proof. Let G be a digraph, and T ⊆ V (G), r ∈ V (G) \ T . Though DiSTP

problem optimizes over the number of edges (recall unit cost!) of a Steiner treeS ⊆ G rooted from r and spanning T , there is a simple equality; |E(S)| =|V (S)| − 1. Hence we can, instead, minimize the cardinality of X = V (S) suchthat X induces in G directed paths from r to all vertices of T .

Similarly to Lemma 5.14, we can thus write (with constants r and T )

∀t ∈ T ∀Z ⊆ X(

(r ∈ Z ∧ t 6∈ Z) → ∃x ∈ Z, y ∈ X\Z arc(x, y))

. ⊓⊔

20

Proofs for Section 3.3 (MaxDiCut)

Theorem 3.6. The MaxDiCut problem is NP-hard on a digraph G even if Gis restricted to be acyclic (implying directed tree-width, DAG-width and Kelly-width, and cycle rank 1) of K-width 4608 and DAG-depth 11.

Proof. To verify that the digraph which is the result of the [LKM08] reductionfrom not-all-equal (NAE) 3SAT has bounded DAG-depth and K-width, we needto slightly modify the construction.

First we may assume the the input instance φ of NAE-3SAT contains noclause with both positive and negative occurrence of the same variable. If this isnot so, we can remove all such clauses, as they are always satisfied in the NAE-SAT sense. Moreover, we can also assume that each variable occurs at most 4times in the input formula. If not, we can replace the k different occurrences ofa variable x with k fresh variables x1, . . . , xk and add the following clauses tothe formula: (x1∨¬x2∨¬x2)∧ (x2 ∨¬x3∨¬x3)∧ . . .∧ (xk ∨¬x1∨¬x1). It is nothard to see that the new formula is satisfied (in the NAE sense) iff the originalformula was satisfied, and every variable occurs at most four times. Moreover,the size of this new formula is linear in the size of φ.

al bixl

¬xl

ci,1,1

ci,1,2

ci,1,3

ci,2,1

ci,2,2

ci,2,3

ci,3,1

ci,3,2

ci,3,3

6|xl|

6|xl|

2

2

2

Fig. 2. MaxDiCut reduction gadget

21

Fig. 2 shows a part of the resulting graph for formula φ, a variable xl and aclause ci, which contains a positive occurrence of xl on the second position. Thelabels on the edges show the weight of the given edge (unlabelled edges haveweight 1). |xl| is the number of occurrences of the variable xl in the formula φ(at most four, as argued above).

To obtain an unweighted graph we use the construction from [LKM08, Theo-rem 3]. This construction first replaces each edge (u, v) of weight k by k paralleledges, and then replaces each parallel edge by a path of length 3 (two freshvertices are added for each such edge).

To compute K-width we notice that the highest number of paths betweensome ai and bj and can be at most 6|xi| ∗ 6|xi| ∗ 2(1 + 2 + 1) ≤ 4608, since|xi| ≤ 4. Finally, the DAG-depth is bounded by the length of longest path,which is 3 + 3 + 3 + 1 + 1 = 11. ⊓⊔

Theorem 3.7. The unweighted MaxDiCut problem on a digraph G of bi-rank-width t is polynomially solvable for every fixed t (i.e. it belongs to the class XP).

Proof. (sketch) We give an XP dynamic programming algorithm running on abi-rank-width parse tree (cf. the appendix of Section 2.1 for the terminology).

We use shortcut notation arcs(G;V0, V1) = { (u, v) ∈ E(G) : u ∈ V0, v ∈ V1 }.Given two t-labeled digraphs G1, G2 and mappings ϕi : V (Gi) → {0, 1} wherei = 1, 2 (here ϕi gives a partition of V (Gi) into V0 = ϕ−1

i (0) and V1 = ϕ−1i (1) ),

we define an equivalence relation: (G1, ϕ1) ≈ (G2, ϕ2) if, and only if, the followingholds for all t-bi-labeled digraphs H and all mappings ψ : V (H) → {0, 1}

∣

∣

∣arcs

(

G1 ⊗ H; ϕ−11 (0), ψ−1(1)

)∣

∣

∣+

∣

∣

∣arcs

(

G1 ⊗ H; ψ−1(0), ϕ−11 (1)

)∣

∣

∣=

=∣

∣

∣arcs

(

G2 ⊗ H; ϕ−12 (0), ψ−1(1)

)∣

∣

∣+

∣

∣

∣arcs

(

G2 ⊗ H; ψ−1(0), ϕ−12 (1)

)∣

∣

∣.

In informal words, the relation ≈ captures “all necessary information from G1”needed to find an optimal solution to MaxDiCut on any (bigger) G1 ⊗ H.

Let T be a t-bi-labeling (bi-rank-width) parse tree of the input digraph G,constructed from Theorem 2.5 (Lemma 5.3). Our algorithm processes T inthe leaves to root direction. At every node s of T , parsing a t-labeled sub-digraph Gs, and for every equivalence class C of ≈, we remember a mappingϕ : V (Gs) → {0, 1} achieving maximum cardinality of arcs

(

Gs;ϕ−1(0), ϕ−1(1)

)

among all (Gs, ϕ) ∈ C. This information can be easily combined from the twodescendants in our parse tree processing. The maximum value (over all classesof ≈) recorded at the root of T is then the optimal solution.

It remains to bound the number of classes of ≈. From the definition of t-bi-labeling join operator ⊗ (cf. the appendix of Section 2.1), we straightforwardlyderive the following claim: Let a signature of (G, ϕ), where G = (G, lab) is at-labeled digraph, be the pair of multisets 〈{ lab(x) : x ∈ ϕ−1(0) }, { lab(x) : x ∈ϕ−1(1) }〉. If (G1, ϕ1) and (G2, ϕ2) have the same signature, then (G1, ϕ1) ≈(G2, ϕ2).

22

The total number of signatures for t-labeled n-vertex digraphs is clearly atmost n2·2t

since every lab(x) ∈ GF (2)t and we record the multiplicities of all

labels. Hence our above outlined algorithm runs in time nO(2t) which is polyno-mial in n for every fixed t. ⊓⊔

Proofs for Section 3.4 (OCN)

Theorem 3.8. The problem (4-OCN) to decide whether a digraph G satisfiesχo(G) ≤ 4 is NP-complete even if G is acyclic of K-width 3 and DAG-depth 5.

Proof. We use the following easy claim from [CD06] as the starting point of ourreduction: Let R be the digraph on the right-hand side of Fig. 3, and Q be theacyclic digraph on the left-hand side. Then every oriented 4-colouring of Q mustinduce a homomorphism into R such that b is mapped to B and f1, f2 are bothmapped to F .

Qf1

b

f2

→

FB

A T

R

Fig. 3. Forcing a 4-colouring homomorphism

We reduce from NP-complete not-all-equal (NAE) 3SAT problem, which hasan input CNF formula ϕ with exactly three literals in each clause, and thequestion is whether ϕ has a satisfying assignment such that no clause receivesthree times true. We replace each variable x of ϕ with a gadget depicted in Fig. 4left, consisting of a copy of Q, two arcs leaving the copy of vertex b into newvertices p and n, a new path of length 5 from p to n, and the necessary numberof terminals for the x and ¬x literals occurring in ϕ, each adjacent from p orn, respectively. Then we replace each clause C of ϕ with a gadget depicted inFig. 4 right, consisting of three directed paths of lengths 3, 4, 5, with a commonsource. The ends of these paths are the terminals for the literals of C.

Let Gϕ be the digraph obtained from all these variable and clause gadgets(pairwise disjoint so far) by identifying all the pairs of corresponding (in ϕ)literal terminals. We claim that ϕ is NAE satisfiable if and only if the orientedchromatic number of Gϕ is 4. It follows from the following sequence of claims:

– Any oriented 4-colouring of Gϕ is a homomorphism into the above digraph R.– In any homomorphism of the variable gadget into R, the vertices p, n are

mapped into {A,T}. Furthermore, the colours of p and n must be distinct.

23

b

p

n

x

x

¬x

¬x

ℓ1

ℓ2

ℓ3

C

Fig. 4. Variable and clause gadgets for 4-OCN reduction

Hence all the x-terminals of the gadget are mapped to T and all the ¬x-terminals are mapped to F (meaning x is valued true), or vice versa (meaningx is valued false).

– A simple case-analysis shows that any homomorphism of the clause gadgetinto R such that ℓ1, ℓ2, ℓ3 are mapped into {T, F} has an additional propertythat not all three colours of ℓ1, ℓ2, ℓ3 are the same (meaning that this clauseis NAE satisfied).

– On the other hand, for both possible surjective mappings p : {x,¬x} →{T, F} there exist homomorphisms of the variable gadget into R extend-ing p. Similarly for all surjective mappings q : {ℓ1, ℓ2, ℓ3} → {T, F} thereexist homomorphisms of the clause gadget into R extending q.

Secondly, we claim that Gϕ has K-width 3 and DAG-depth 5. Since all theterminals in our construction of the acyclic digraph Gϕ are sinks, it is enoughto verify the claimed properties for each gadget separately. The K-width boundis easy; we get up to three distinct paths between two vertices in a copy of Q(Fig. 3). We now show a winning strategy for the cops on a variable gadget in5 moves. In the first two moves, cops land on b and p, and then the robber iseasily caught on one of the remaining directed paths of length ≤ 6. For a clausegadget, just 4 moves suffice when the first cop lands on C. ⊓⊔

Proposition 3.9. The problem (c-OCN) to decide χo(G) ≤ c on an input di-graph G of bi-rank-width t is fixed parameter tractable with parameters c and t.

Proof. We write an MSO1 formula

∃X1, . . . ,Xc

∧

i=1,...,c

∀x, y ∈ Xi

(

¬arc(x, y))

∧∧

i,j=1,...,c

∀x, y ∈ Xi, z, t ∈ Xj

(

arc(x, z) → ¬arc(t, y))

which describes valid oriented colouring classes X1, . . . Xc in a graph G. Hencethe result follows from Theorem 3.6. ⊓⊔

24

At last we remark that, although there is an XP algorithm computing thechromatic number of a given graph of bounded rank-width, it is open whethersuch an algorithm could exist for computing the oriented chromatic numberof a digraph of bounded bi-rank-width. It seems that the known “undirected”algorithm does not extend in this way.

Proofs for Section 3.6 (φ-MSO1mc)

Theorem 3.12 (cf. [CMR00], and [Kan08,GH08]).Every LinEMSO1 optimization problem is fixed parameter tractable when re-stricted to digraphs of bi-rank-width t, with a parameter t.

Proof. (sketch) Given an input digraph G of bi-rank-width t, we first use Theo-rem 2.5 to compute a width-t bi-rank-decomposition of G, and then construct anexpression XG of clique-width ≤ 2t+1 − 1 for G using [Kan08, Proposition 5.3].Now, although the formulations in [CMR00] speak only about FPT solvabilityof LinEMSO1 problems on undirected graphs (their τ1 graph structure has asymmetric adjacency relation) of bounded clique-width, there is no apparentmathematical reason why not to extend the whole interpretation scheme thereto digraphs. Therefore, [CMR00] (indirectly) implies our theorem.

Alternative proof. Nevertheless, the indirect interpretability approach (based on[CMR00]) to Theorem 3.12 has some disadvantages. First, there is no apparentexplicit algorithm behind it, and no “nice” estimate of run-time dependency ont for particular problems except a generic “tower of exponents”. Second, theclique-width parameter in the above reduction may grow up to exponentially int which is not good in applications.

We propose another, more explicit approach to proving Theorem 3.12, basedon the bi-labeling parse trees of [Kan08] and the proof method of [GH08, The-orem 4.2] (which has been an alternative to [CMR00] on undirected graphs ofbounded rank-width) Given an input digraph G of bi-rank-width t, we first useTheorem 2.5 to compute a width-t bi-rank-decomposition of G, and then wetranslate this decomposition into a t-bi-labeling parse tree, e.g. in quadratictime using the method of [GH08, Theorem 2.2].

Now, with the same “automata–regularity” tools as used in [GH08, Theo-rem 4.2], we prove (constructively) the following: For every MSO1 formula ϕand fixed t, there is a finite tree automaton accepting exactly those t-bi-labelingparse trees T giving a digraph GT such that GT |= ϕ (when ϕ has free variables,we naturally consider GT equipped with interpretations for these free variables).

In a dynamic processing of the input labeling parse tree, we then keep trackonly of suitable “optimal” representatives of all possible interpretations of thefree variables in ϕ, indexed by the states of the automaton. The overall runningtime is O(f(t) ·n3) for some computable f depending on the problem (on ϕ). ⊓⊔

Proposition 3.13. The MSO1 model checking problem is NP-hard even whenrestricted to acyclic digraphs of K-width 1 and DAG-depth 2.

25

v1v2

wv1,1wv2,1

wv1,2wv2,2

wv1,3wv2,3

v3v4

wv3,1wv4,1

wv3,2wv4,2

wv3,3wv4,3

v5v6

wv5,1wv6,1

wv5,2wv6,2

wv5,3wv6,3

v7v8

wv7,1wv8,1

wv7,2wv8,2

wv7,3wv8,3

· · ·· · ·

S

T

Fig. 5. The construction used in Theorem 3.14

Proof. We show an MSO1 interpretation of undirected graphs in suitable di-graphs. Given a graph H without isolated vertices, we construct an acyclic di-graph G of K-width 1 and DAG-depth 2: For every edge e = uv of H, we add anew vertex xe and replace e with two arcs uxe, vxe. Notice that G has no directedpath on 3 vertices. A vertex v of H can be then identified in G with ∃x(arc(v, x),and a predicate edge(u, v) can be written as ∃x(arc(u, x)∧ arc(v, x)). The claimfollows since MSO1 model checking is NP-hard on undirected graphs. ⊓⊔

Proofs for Section 3.7 (φ-LTLmc)

We briefly fix the LTL-notation used in the following proofs. Atomic propertiesdo hold on vertices (states). The Boolean connectors are as usual, the operatorsnext and eventually (in the future) are denoted by capital letters X and F . Weassume that if a run reaches a sink, it repeats its symbol infinitely often to avoidthe existence of finite runs. For clarity, this will be made explicit with self-loopsin the figures.

Theorem 3.14. The φ-LTLmc problem is coNP-hard even when the input di-graph is restricted to have K-width 1, DAG-depth 4, and bi-rank-width 2.

Proof. We use a reduction from the DS (undirected dominating set) problem,which is a folklore NP-complete problem even when the input is restricted tocubic graphs. Let G = (V,E), k ∈ N be an input instance for DS such that thegraph G has all degrees 3. We construct the following instance of φ-LTLmc withan underlying digraph G′ = (V ′, E′):

26

v1

wv1,1 wv1,2

wv1,3 v2

wv2,1 wv2,2

wv2,3 v3

wv3,1 wv3,2

wv3,3 v4

wv4,1 wv4,2

wv4,3. . .

TS

Fig. 6. The construction used in Theorem 3.15

For each v ∈ V , we define V ′v = {uv , uv,1, uv,2, uv,3}. Then V ′ = {s, t, c} ∪

⋃

v∈V V′v . We let properties S hold in s, and T in t. If the neighbours of each v in

G are wv,1, wv,2, wv,3, then let v hold in uv , and wv,i hold in uv,i for i = 1, 2, 3.Edges are added as follows.

E′ = {(s, c), (c, t)} ∪ { (c, uv), (uv,3, c) : v ∈ V }

∪ { (uv , uv,1), (uv,1, uv,2), (uv,2, uv,3), : v ∈ V }.

ThenF = ¬ (X1+5k+1 T ∧

∧

v∈VFv)

holds on G′ iff G does not contain a dominating set of size k. See Figure 5 foran illustration.

Assume there is any dominating set D of size k in G. Then the run (i.e. di-rected walk in G′) R starting in s, following the cycle through each V ′

v ∪ {c} forall v ∈ D, and visiting only t afterwards, does not satisfy F : After 1 + 5k + 1steps, the sink t is reached (where T holds), and since C is a dominating set,each v ∈ V holds at some point in P – namely when V ′

w is traversed for somew ∈ D that dominates v.

Now assume that G contains no dominating set of size k. Then, no run R oflength at most 1 + 5k can satisfy

∧

v∈V Fv : Since at most k sets V ′v1, . . . , V ′

vk

are visited by R and since the corresponding nodes v1, . . . , vk ∈ V cannot forma dominating set in G, there is some w ∈ V such that w does not hold on R.Hence, any path that satisfies

∧

v∈V Fv cannot satisfy X1+5k+1 t at the sametime. Therefore, F holds on G′.

Finally, the digraph G′ has clearly K-width 1, and it can be shown to haveDAG-depth 4. To prove that the bi-rank-width of G′ is at most 2 (recall Sec-tion 2.1), we look at the set Y = {s, t, c}. Then both matrices A+

Y and A−Y have

only one nonzero row each, and so brkG′(Y ) = 2. Furthermore, the subdigraphG′−Y is a collection of paths, and so brkG′ can easily be 2-branched on V ′ \Y .Hence the bi-rank-width of G′ is at most 2. ⊓⊔

Theorem 3.15. LTL model checking is coNP-complete on DAGs.

Proof. For containment in coNP, note that the NTM just has to guess a run onwhich the formula does not hold. Since there are no loops (except self-loops),the length of this run is polynomially bounded.

27

For the hardness part, we use a similar construction to the one used in The-orem 3.14. See in Figure 6. Let G = (V,E), k ∈ N be an input instance for DSsuch that the graph G has all degrees 3. We construct the following instance ofφ-LTLmc with an underlying digraph G′ = (V ′, E′):

Again, V ′v = {uv , uv,1, uv,2, uv,3} for each v ∈ V . Then V ′ = {s0, s1, . . . , s|V |,

t}∪⋃

v∈V V′v . We let S hold in s0 and T hold in t. If the neighbours of each v in

G are wv,1, wv,2, wv,3, then let v hold in uv , and wv,i hold in uv,i for i = 1, 2, 3.Edges are added as follows, assuming V = {v1, . . . , vn} (in arbitrary order).

E′ = { (si−1, si) : 1 ≤ i ≤ |V | } ∪ {(s|V |, t)} ∪ { (si−1, uvi), (uvi,3, si) : 1 ≤ i ≤ |V | }

∪ {(uvi, uvi,1), (uvi ,1, uvi,2), (uvi,2, uvi,3) : 1 ≤ i ≤ |V |}

ThenF = ¬

(

X4k+|V | t ∧∧

v∈VFv

)

holds on G′ iff G does not contain a dominating set of size k. The rest followsin the same way as in the proof of Theorem 3.14. ⊓⊔

Proofs for Section 4 (DFVS)

Theorem 4.1. If a digraph G is given with a directed feedback vertex set ofsize k, then the Kernel problem can be solved in time O(2k · |V (G)|2).

Proof. (sketch) We are actually going to solve the annotated Kernel problem,in which the input is a digraph G and a set U ⊆ V (G), and the task is to find akernel which is a subset of U .

For a given DAG G and set U , we easily solve annotated Kernel using thefollowing reduction rules:

– The setK ⊆ V (G) of all sink vertices ofG (which is acyclic) must be includedin the kernel. If K 6⊆ U , then no such kernel exists.

– Hence all the in-neighbours N−(K) already have an arc into the kernel, andso N−(K) can be ignored further on. We call the procedure recursively onG− (K ∪N−(K)) and U \ (K ∪N−(K)).

We now consider an arbitrary digraph G with a feedback set S ⊆ V (G) ofsize k, and solve annotated Kernel for G and U ⊆ V (G) as follows. We cyclethrough all 2k subsets Z ⊆ S, looking for a kernel that intersects S in Z:

– If Z is not independent in G, then this iteration fails.– We include Z into the kernel. Hence all the in-neighbours N−(K) already

have an arc into the kernel, and so N−(K) can be ignored further on. All theout-neighbours N+(K) must stay out of the kernel, and so they can simplybe removed from the set U .

– Since all the vertices of S \ Z are not in the kernel, the incoming edges ofS \ Z have no influence on the problem, and so they can be removed fromthe digraph G, making a new digraph G′.

28

– Finally, the digraph G′−Z is acyclic. Therefore, we call the previous proce-dure on G′ − (Z ∪N−(K)) and U \ (Z ∪N−(K) ∪N+(K)).

It remains to straightforwardly verify that this procedure gives a correct answer.

29